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    NOVA Eienstien

    Just watched NOVA today. A good portrait of AE and his evolution of relativity.

    Apparently he collaborated with a mathematician on the math he required and Hilbert coopted his ideas and tried to beat him.

    At the end he went through four weeks of peer review presentations finding mistakes and correcting until he had the final theory.

    What stood out to me was all he would have had was a slide rule, a mechanical calculator(they existed), and paper and pencil. His calculations that showed an accurate prediction of the orbit of Mercury were all done by hand. He made mistakes, some as simple as copying or writing down an equation incorrectly. Imagine if he had a simple 80s era programmable scientific calculator with a printer.

    For those who want to see how science works from thought to equations to demonstrations watch the show. Nobody works in a vacuum. Trial and error is part of the process.

    What was also interesting was a British scientist did the observation that showed mass deflects light. In the day it was a big deal, a British and German scientistcalaborating.

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    Administrator lpetrich's Avatar
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    Quote Originally Posted by steve_bank View Post
    Imagine if he had a simple 80s era programmable scientific calculator with a printer.
    Or any full-scale computer from the 1950's to the present. What would be the most useful for him would, I think, be computer-algebra software, and the first such software was developed in the 1960's. It is now very well-developed, with both proprietary and open-source CAS packages available.

    He would have loved using CAS to do the math of general relativity -- differential geometry is nightmarishly complicated. For my part, it took some time to write the differential-geometry code itself, but once it was written, it is easy to plug in whatever space geometry I want.

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    The biggest moments in Physics occurred generally without electronic calculators. It is amazing how much was discovered from much broader observations and paper / pencil.

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    Yes, it was necessary to do a lot of calculations by hand, especially in astronomical work. Astronomers and mathematicians and the like often hired assistants, called "computers" back then, to do much of the work. To reduce the amount of computation labor needed, they often calculated tables of mathematical functions and positions of celestial bodies and the like.

    We may know about using logarithm tables for doing multiplication, but did any of you know that trigonometric tables were once used for that purpose? Around 1600, Prosthaphaeresis ("addition and subtraction") was often used. It uses trigonometric identities

    cos(a)*cos(b) = (1/2)*(cos(a-b) + cos(a+b))
    sin(a)*sin(b) = (1/2)*(cos(a-b) - cos(a+b))
    sin(a)*cos(b) = (1/2)*(sin(a-b) + sin(a+b))

    for that.

    Half a century later, logarithms became common, and they were often used for doing multiplication until the late twentieth century:

    log(x*y) = log(x) + log(y)

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    Quote Originally Posted by lpetrich View Post
    ... snip ...

    We may know about using logarithm tables for doing multiplication, but did any of you know that trigonometric tables were once used for that purpose?

    ... snip ...
    Indeed so, I still have my "CRC Tables" (a book of all sorts of math tables) all physics and engineering students lived by when more precision was needed than the slide rule afforded. For even more precision, it was necessary to interpolate for values between those listed in the tables. I still have a few of my old slide rules but I never got one of those twenty inch models used by chemistry majors.

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    Astronomical tables or ephemerides (Ephemeris) have been calculated for millennia.

    Historian of Science Owen Gingerich has researched some medieval and early-modern ephemerides in "Crisis" versus Aesthetic in the Copernican Revolution
    I am willing to grant that Copernicus' cosmology represents, in a certain profound sense, a simplification, but I refuse to concede that the Ptolemaic theory had by the beginning of the sixteenth century reached a complex, patched-up state nearing collapse. In terms of the detailed mechanism for any particular planet, it would have been very difficult for Copernicus' contemporaries to distinguish between the two schemes on the basis of complexity.
    OG checked these ephemerides by comparing them to planet positions extrapolated from recent observations. The ones he looked at had two characteristic patterns of errors, one for using Ptolemy's model and one for using Copernicus's model. I use "model" here to represent not only the overall picture but also the calculation details.

    So the piling up of epicycles had not happened. Pre-Copernican calculations giving close to the same pattern of errors means that they were following Ptolemy rather closely.

    The two models had similar complexities and similar sizes of errors. Copernicus's model had the advantage of greater simplicity, and it also enabled calculating the position angles and periods of Ptolemy's outer-planet epicycles. So the theory needed fewer parameters.


    The idea of adding more and more epicycles seems like something relatively easy to do over the last century, especially with our understanding of trigonometric series. But there apparently wasn't much of that in the late Middle Ages and early modern times.

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    Johannes Kepler (1571-1630) is an interesting case. He wanted to analyze Tycho Brahe's observations but TB only gave JK TB's Mars observations. That was a difficult problem -- Mars was sometimes off from existing calculations by as much as 5d. He nevertheless succeeded, going through some 40 shapes before settling on an ellipse with the Sun at one focus.

    Along the way, he fixed Copernicus's theory of Mars's orbit plane. Copernicus had included a wobble in the inclination of Mars's orbit with respect to the Earth's, but Kepler discovered that it was due to Copernicus using the "mean Sun" as his Sun reference. That is the averaged-out direction of the Sun relative to the Earth. When he put in the Sun's true direction, this wobble disappeared. Copernicus didn't know his own riches, he wrote about that.

    All this work took several years, but it would have been much quicker with present-day computers, or even those of half a century ago.

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    Sir Isaac Newton found himself in a similar situation. He was very successful in solving the gravitational two-body problem, but more objects were much more difficult. In particular, he concerned himself with the motions of the Moon. Its orbit is inclined to the Earth's orbit by about 5 degrees, and its orbit eccentricity is about 0.055. Both its line of nodes and its line of apsides precess, the line of nodes going backward and the line of apsides going forward. The line of nodes is between the ascending node, where the Moon crosses northward, and the descending node, where the Moon crosses southward. The line of apsides is between the closest and farthest distances, the periapsis and apoapsis, the pericenter and apocenter, and for orbiting the Earth, the perigee and apogee. In addition to these effects, the Moon has plenty of smaller wobbles in its motion, the largest of which are the "evection" (about 1.3 degrees) and the "variation" (about 0.7 degrees).

    When Newton tried to calculate these effects, he ran into a problem. He could not solve the complete three-body problem, and it has continued to be an unsolved problem. So he calculated the Moon's orbit as the Sun's perturbations of the Moon's orbiting the Earth. He did it bytreating the Sun's effects as a small departure from the unperturbed orbit, and one can repeat this operation using the orbit with the previously-found amount of permutation. Newton himself did not do that, though his successors did.

    Newton found a precession rate of (3/4)*we^/wm where we is the Earth's orbit rate and wm is the Moon's. Let us look at the numbers.

    What Period
    Calculated 17.83
    Nodes 18.60
    Apsides 8.85
    The periods are in (Earth) years. Newton's calculation was close to the observed rate for the nodes, but it was only half the observed rate for the apsides. This discrepancy, Newton told us, "made his head ache".

    A little over half a century later, a certain Alexis Clairaut took on this problem, and he carried his calculations further than Newton had done, and he resolved that discrepancy. Nothing new was needed, only an improved calculation.

    But Newton and Clairaut had to do their calculations by hand, while we can nowadays use computer-algebra software to do so.

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